molecules Article Spin Density Topology Giovanna Bruno 1 , Giovanni Macetti 2 , Leonardo Lo Presti 1 and Carlo Gatti 3,4,* 1 Dipartimento di Chimica, Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy; [email protected] (G.B.); [email protected] (L.L.P.) 2 Laboratoire de Physique et Chimie Théoriques (LPCT), Université de Lorraine & CNRS, 1 Boulevard Arago, F–57078 Metz, France; [email protected] 3 CNR–SCITEC, Istituto di Scienze e Tecnologie Chimiche sezione di via Golgi, c/o Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy 4 Istituto Lombardo, Accademia di Scienze e Lettere, via Brera 28, 20100 Milano, Italy * Correspondence: [email protected] Academic Editor: Ángel Martín Pendás Received: 30 June 2020; Accepted: 1 August 2020; Published: 2 August 2020 Abstract: Despite its role in spin density functional theory and it being the basic observable for describing and understanding magnetic phenomena, few studies have appeared on the electron spin density subtleties thus far. A systematic full topological analysis of this function is lacking, seemingly in contrast to the blossoming in the last 20 years of many studies on the topological features of other scalar fields of chemical interest. We aim to fill this gap by unveiling the kind of information hidden in the spin density distribution that only its topology can disclose. The significance of the spin density critical points, the 18 different ways in which they can be realized and the peculiar topological constraints on their number and kind, arising from the presence of positive and negative spin density regions, is addressed. The notion of molecular spin graphs, spin maxima (minima) joining paths, spin basins and of their valence is introduced. We show that two kinds of structures are associated with a spin–polarized molecule: the usual one, defined through the electron density gradient, and the magnetic structure, defined through the spin density gradient and composed in general by at least two independent spin graphs, related to spin density maxima and minima. Several descriptors, such as the spin polarization index, are introduced to characterize the properties of spin density critical points and basins. The study on the general features of the spin density topology is 3 followed by the specific example of the water molecule in the B1 triplet state, using spin density distributions of increasing accuracy. Keywords: spin density; topology; quantum chemical topology; spin density critical points; spin maxima and minima joining paths; molecular spin graph; spin density basins; spin density source 3 function; water B1 triplet 1. Introduction 1.1. Introductory Remarks and Scope The electron density ρ is a sum of two components due to the α– and the β–spin electrons. When ρα and ρβ are not equal, there is spin polarization and the spin density s, given by their difference, is not zero [1] (p. 35). The spin density s and the electron density ρ are the two 3D space functions acting as basic variables in spin density functional theory (SDFT), which is the required generalization of the DFT in the presence of a magnetic field, besides the usual scalar external potential due to the nuclei. Even when this additional potential is absent, SDFT is of importance as “it allows to build more physics into the approximate exchange–correlation functional through its spin–dependence” [1] (p. 169). As a Molecules 2020, 25, 3537; doi:10.3390/molecules25153537 www.mdpi.com/journal/molecules Molecules 2020, 25, 3537 2 of 26 result, a more correct treatment of spin–polarized systems, such as open–shell atoms and molecules or molecules near the dissociation limit, is possible [1]. Apart its function in SDFT, the spin density reflects the interactions that are taking place among the electron spins within a system and so also plays a fundamental role for understanding magnetic phenomena [2–6]. For example, the detailed knowledge of the spin density distribution in a molecule is essential for interpreting how spin polarization propagates in molecular complexes and crystals, allowing one to highlight and rationalize the various magnetic interactions as a function of molecular orientation and packing [4,7,8]. These interactions are comparable or even smaller in energy than the thermal energy, kBT, yet they can be exploited for designing magnetic materials with foreseeably important technological applications [3]. Despite the importance of the electron spin density, only a few studies have appeared on its subtleties thus far. We are not aware of a systematic topological analysis of this function, seemingly in contrast with the blossoming in the last 20 years of many topological analyses of scalar fields related to chemistry and defined in terms of quantum mechanical functions [9–12]. Among these studies, which are collectively gathered under the umbrella of Quantum Chemical Topology [9,10], a dominant place is undoubtedly taken by those on the electron density, the spin density companion key ingredient function in SDFT. Such a prominent position is justified by the cornerstone role that the electron density plays in the Bader’s Quantum Theory of Atoms in Molecules (QTAIM) [13]. In QTAIM, all elements of chemical structure—atoms, bonds, rings, cages and the structure itself—can be rigorously defined through the electron density topology. The aim of the present work is to unveil the information hidden in the spin density distribution that only its topology can disclose. Questions such as the meaning of the spin density critical points (where s = 0), the physical conditions for their occurrence, the different ways they may realize r depending on their kind (maxima, saddles, minima) and on the relative dilution or concentration of the spin majority and the spin minority distributions at the critical point, are first addressed. Then the notions of spin graphs, spin molecular structure, spin maxima and minima joining paths, spin basins and their valence are introduced. The study is organized into two intimately connected parts: the first one is strictly general and of a fundamental nature (Section2), while the second one is related to the 3 exemplary case of the water molecule in the B1 state and worked out in very much detail (Section3). It is shown that a spin–polarized molecule, at variance with a non–polarized one, is described through two kinds of structures. One is customarily defined through the electron density gradient and is in a one–to–one correspondence with the empirical notion of chemical structure [13]. The other one, the magnetic structure, is obtained through the s field and is in general composed by at least two r independent spin graphs, associated with the spin density maxima and minima, respectively. While the conventional chemical structure and the spin density maxima magnetic structure are quite robust against changes in the way they are computed, the spin density minima magnetic structure is instead very much dependent on the way it is evaluated. This fact also has a clear impact on the number of recovered critical points and on the way the Poincaré–Hopf theorem [13–16], which expresses the topological constraint among the numbers of each kind of critical points, is separately satisfied in the positive and negative spin density regions. Properties of spin density basins, defined in two possible alternative ways, are discussed and some interesting indexes to characterize their spin polarization nature are proposed. The relative capability of the various spin basins to determine the spin density magnitude and sign in the various molecular regions is also explored, using the source function tool (see infra) [17–19]. The present study suggests that the spin density topology can disclose a wealth of interesting information. It so surely deserves further studies both on its fundamental aspects and on the exploration of its features in more complex systems. We are currently performing a few studies along these lines, using spin–polarized transition metal complexes as more difficult and, possibly, interesting cases. A next step will then involve the spin density topology of interacting complexes, like those studied in [4–8]. Molecules 2020, 25, 3537 3 of 26 1.2. Literature Context Before presenting our work on the spin density topology, we deem it important to briefly recall a few significant studies, which inspired us and introduce some terminology and useful concepts. All these studies were similarly aimed at disclosing important and general details of spin–polarized distributions as well as of the mechanisms of spin propagation from a paramagnetic center to the neighboring atoms or ligands or from a molecule to other molecules in a molecular complex or solid. In their 2005 review paper, Ruiz et al. [4] discussed the characteristic shapes of the spin density distributions around a transition metal atom, as a function of its electronic configuration. This analysis represents a first qualitative study of the topology of the spin density in metal coordination complexes. Ruiz et al. also summarized the characteristics trends involving the two main mechanisms through which an α– unpaired electron in a singly occupied molecular orbital (SOMO) can place some spin density at the other atoms of the molecule and at its own other atomic orbitals. One mechanism is spin delocalization. It is due to the contribution of the various atomic orbitals to the SOMO and results in a distribution of positive spin density throughout the molecule. The other one is spin polarization, through which the positive α spin at a center may induce some spin density of the opposite sign at the atoms bonded to it. Spin polarization is due to electron exchange, which reduces the electron repulsion between electrons of the same–spin and favors their proximity relative to electrons of opposite spin. As a result, the α–electron of a bonding pair of electrons will preferentially stay closer to the atom of the bonded pair of atoms with higher excess of positive spin, while a corresponding β–spin polarization will be induced on the remaining atom.